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Properties

Label 7650.2.a.c.1.1
Level $7650$
Weight $2$
Character 7650.1
Self dual yes
Analytic conductor $61.086$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7650,2,Mod(1,7650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7650.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.0855575463\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2550)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7650.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{7} -1.00000 q^{8} -2.00000 q^{11} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +8.00000 q^{19} +2.00000 q^{22} +1.00000 q^{23} +2.00000 q^{26} -4.00000 q^{28} +4.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{34} -3.00000 q^{37} -8.00000 q^{38} +1.00000 q^{41} +6.00000 q^{43} -2.00000 q^{44} -1.00000 q^{46} +4.00000 q^{47} +9.00000 q^{49} -2.00000 q^{52} -13.0000 q^{53} +4.00000 q^{56} -4.00000 q^{58} -15.0000 q^{59} +5.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +10.0000 q^{67} -1.00000 q^{68} +1.00000 q^{71} +16.0000 q^{73} +3.00000 q^{74} +8.00000 q^{76} +8.00000 q^{77} +12.0000 q^{79} -1.00000 q^{82} -11.0000 q^{83} -6.00000 q^{86} +2.00000 q^{88} +2.00000 q^{89} +8.00000 q^{91} +1.00000 q^{92} -4.00000 q^{94} -9.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514 0.104257 − 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 − 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −13.0000 −1.78569 −0.892844 − 0.450367i \(-0.851293\pi\)
−0.892844 + 0.450367i \(0.851293\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −15.0000 −1.95283 −0.976417 − 0.215894i \(-0.930733\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 − 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 − 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 0 0
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.00000 −0.110432
\(83\) −11.0000 −1.20741 −0.603703 − 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −3.00000 −0.295599 −0.147799 − 0.989017i \(-0.547219\pi\)
−0.147799 + 0.989017i \(0.547219\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 13.0000 1.26267
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −15.0000 −1.41108 −0.705541 − 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) 15.0000 1.38086
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −5.00000 −0.452679
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) −32.0000 −2.77475
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) 1.00000 0.0848189 0.0424094 − 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.00000 −0.0839181
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 0 0
\(148\) −3.00000 −0.246598
\(149\) −1.00000 −0.0819232 −0.0409616 − 0.999161i \(-0.513042\pi\)
−0.0409616 + 0.999161i \(0.513042\pi\)
\(150\) 0 0
\(151\) 1.00000 0.0813788 0.0406894 − 0.999172i \(-0.487045\pi\)
0.0406894 + 0.999172i \(0.487045\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −12.0000 −0.954669
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 7.00000 0.548282 0.274141 − 0.961689i \(-0.411606\pi\)
0.274141 + 0.961689i \(0.411606\pi\)
\(164\) 1.00000 0.0780869
\(165\) 0 0
\(166\) 11.0000 0.853766
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) 8.00000 0.608229 0.304114 − 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) −9.00000 −0.672692 −0.336346 − 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 − 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 − 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) −24.0000 −1.72756 −0.863779 − 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 2.00000 0.142494 0.0712470 − 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 − 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) −16.0000 −1.12298
\(204\) 0 0
\(205\) 0 0
\(206\) 3.00000 0.209020
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −13.0000 −0.892844
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) −17.0000 −1.13840 −0.569202 − 0.822198i \(-0.692748\pi\)
−0.569202 + 0.822198i \(0.692748\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 15.0000 0.997785
\(227\) −22.0000 −1.46019 −0.730096 − 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 − 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 3.00000 0.196537 0.0982683 − 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −15.0000 −0.976417
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) −6.00000 −0.388108 −0.194054 − 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 − 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) 4.00000 0.246651 0.123325 − 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 32.0000 1.96205
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −21.0000 −1.27566 −0.637830 − 0.770178i \(-0.720168\pi\)
−0.637830 + 0.770178i \(0.720168\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −31.0000 −1.86261 −0.931305 − 0.364241i \(-0.881328\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) −1.00000 −0.0599760
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 − 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 7.00000 0.416107 0.208053 − 0.978117i \(-0.433287\pi\)
0.208053 + 0.978117i \(0.433287\pi\)
\(284\) 1.00000 0.0593391
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 16.0000 0.936329
\(293\) −31.0000 −1.81104 −0.905520 − 0.424304i \(-0.860519\pi\)
−0.905520 + 0.424304i \(0.860519\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.00000 0.174371
\(297\) 0 0
\(298\) 1.00000 0.0579284
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) −1.00000 −0.0575435
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) −22.0000 −1.25561 −0.627803 − 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) 0 0
\(311\) −5.00000 −0.283524 −0.141762 − 0.989901i \(-0.545277\pi\)
−0.141762 + 0.989901i \(0.545277\pi\)
\(312\) 0 0
\(313\) −20.0000 −1.13047 −0.565233 − 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 26.0000 1.46031 0.730153 − 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) −7.00000 −0.387694
\(327\) 0 0
\(328\) −1.00000 −0.0552158
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −11.0000 −0.603703
\(333\) 0 0
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 − 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −8.00000 −0.430083
\(347\) 16.0000 0.858925 0.429463 − 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) 16.0000 0.851594 0.425797 − 0.904819i \(-0.359994\pi\)
0.425797 + 0.904819i \(0.359994\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 9.00000 0.475665
\(359\) −36.0000 −1.90001 −0.950004 − 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −11.0000 −0.578147
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) −2.00000 −0.104399 −0.0521996 − 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) 52.0000 2.69971
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −1.00000 −0.0513665 −0.0256833 − 0.999670i \(-0.508176\pi\)
−0.0256833 + 0.999670i \(0.508176\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 10.0000 0.511645
\(383\) −20.0000 −1.02195 −0.510976 − 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 0 0
\(388\) 0 0
\(389\) −9.00000 −0.456318 −0.228159 − 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) −1.00000 −0.0505722
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) 21.0000 1.05396 0.526980 − 0.849878i \(-0.323324\pi\)
0.526980 + 0.849878i \(0.323324\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 − 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 16.0000 0.794067
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 19.0000 0.939490 0.469745 − 0.882802i \(-0.344346\pi\)
0.469745 + 0.882802i \(0.344346\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.00000 −0.147799
\(413\) 60.0000 2.95241
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 16.0000 0.782586
\(419\) −24.0000 −1.17248 −0.586238 − 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −24.0000 −1.16969 −0.584844 − 0.811146i \(-0.698844\pi\)
−0.584844 + 0.811146i \(0.698844\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 13.0000 0.631336
\(425\) 0 0
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 28.0000 1.34871 0.674356 − 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) −12.0000 −0.572729 −0.286364 − 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) −39.0000 −1.85295 −0.926473 − 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 17.0000 0.804973
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) 42.0000 1.98210 0.991051 − 0.133482i \(-0.0426157\pi\)
0.991051 + 0.133482i \(0.0426157\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) −15.0000 −0.705541
\(453\) 0 0
\(454\) 22.0000 1.03251
\(455\) 0 0
\(456\) 0 0
\(457\) 31.0000 1.45012 0.725059 − 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 18.0000 0.841085
\(459\) 0 0
\(460\) 0 0
\(461\) 1.00000 0.0465746 0.0232873 − 0.999729i \(-0.492587\pi\)
0.0232873 + 0.999729i \(0.492587\pi\)
\(462\) 0 0
\(463\) −35.0000 −1.62659 −0.813294 − 0.581853i \(-0.802328\pi\)
−0.813294 + 0.581853i \(0.802328\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −3.00000 −0.138972
\(467\) 3.00000 0.138823 0.0694117 − 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 0 0
\(469\) −40.0000 −1.84703
\(470\) 0 0
\(471\) 0 0
\(472\) 15.0000 0.690431
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) 16.0000 0.731059 0.365529 − 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −5.00000 −0.226339
\(489\) 0 0
\(490\) 0 0
\(491\) −39.0000 −1.76005 −0.880023 − 0.474932i \(-0.842473\pi\)
−0.880023 + 0.474932i \(0.842473\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −41.0000 −1.83541 −0.917706 − 0.397260i \(-0.869961\pi\)
−0.917706 + 0.397260i \(0.869961\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −4.00000 −0.178529
\(503\) −41.0000 −1.82810 −0.914050 − 0.405602i \(-0.867062\pi\)
−0.914050 + 0.405602i \(0.867062\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −64.0000 −2.83119
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) −12.0000 −0.527250
\(519\) 0 0
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 34.0000 1.48672 0.743358 − 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) −32.0000 −1.38738
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) 0 0
\(536\) −10.0000 −0.431934
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 15.0000 0.644900 0.322450 − 0.946586i \(-0.395494\pi\)
0.322450 + 0.946586i \(0.395494\pi\)
\(542\) 21.0000 0.902027
\(543\) 0 0
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) −13.0000 −0.555840 −0.277920 − 0.960604i \(-0.589645\pi\)
−0.277920 + 0.960604i \(0.589645\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) −48.0000 −2.04117
\(554\) 31.0000 1.31706
\(555\) 0 0
\(556\) 1.00000 0.0424094
\(557\) −31.0000 −1.31351 −0.656756 − 0.754103i \(-0.728072\pi\)
−0.656756 + 0.754103i \(0.728072\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) −2.00000 −0.0843649
\(563\) 15.0000 0.632175 0.316087 − 0.948730i \(-0.397631\pi\)
0.316087 + 0.948730i \(0.397631\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −7.00000 −0.294232
\(567\) 0 0
\(568\) −1.00000 −0.0419591
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −41.0000 −1.71580 −0.857898 − 0.513820i \(-0.828230\pi\)
−0.857898 + 0.513820i \(0.828230\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 0 0
\(577\) 33.0000 1.37381 0.686904 − 0.726748i \(-0.258969\pi\)
0.686904 + 0.726748i \(0.258969\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) 44.0000 1.82543
\(582\) 0 0
\(583\) 26.0000 1.07681
\(584\) −16.0000 −0.662085
\(585\) 0 0
\(586\) 31.0000 1.28060
\(587\) 25.0000 1.03186 0.515930 − 0.856631i \(-0.327446\pi\)
0.515930 + 0.856631i \(0.327446\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) −3.00000 −0.123299
\(593\) −12.0000 −0.492781 −0.246390 − 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.00000 −0.0409616
\(597\) 0 0
\(598\) 2.00000 0.0817861
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 6.00000 0.244745 0.122373 − 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) 24.0000 0.978167
\(603\) 0 0
\(604\) 1.00000 0.0406894
\(605\) 0 0
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 − 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 − 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 5.00000 0.200482
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) 0 0
\(626\) 20.0000 0.799361
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 − 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) −12.0000 −0.477334
\(633\) 0 0
\(634\) −26.0000 −1.03259
\(635\) 0 0
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 − 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −29.0000 −1.14365 −0.571824 − 0.820376i \(-0.693764\pi\)
−0.571824 + 0.820376i \(0.693764\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) −38.0000 −1.49393 −0.746967 − 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) 0 0
\(652\) 7.00000 0.274141
\(653\) −12.0000 −0.469596 −0.234798 − 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.00000 0.0390434
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) 40.0000 1.55818 0.779089 − 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) 50.0000 1.94477 0.972387 − 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) 11.0000 0.426883
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) −24.0000 −0.928588
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 − 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −12.0000 −0.461197 −0.230599 − 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) −6.00000 −0.229584 −0.114792 − 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) 26.0000 0.990521
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 − 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 8.00000 0.304114
\(693\) 0 0
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) 0 0
\(697\) −1.00000 −0.0378777
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) 0 0
\(701\) 3.00000 0.113308 0.0566542 − 0.998394i \(-0.481957\pi\)
0.0566542 + 0.998394i \(0.481957\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −16.0000 −0.602168
\(707\) −40.0000 −1.50435
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 − 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −2.00000 −0.0749532
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) −9.00000 −0.336346
\(717\) 0 0
\(718\) 36.0000 1.34351
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 11.0000 0.408812
\(725\) 0 0
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 − 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) −8.00000 −0.296500
\(729\) 0 0
\(730\) 0 0
\(731\) −6.00000 −0.221918
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 − 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −20.0000 −0.736709
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 − 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −52.0000 −1.90898
\(743\) 17.0000 0.623670 0.311835 − 0.950136i \(-0.399056\pi\)
0.311835 + 0.950136i \(0.399056\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 − 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 − 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 1.00000 0.0363216
\(759\) 0 0
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 − 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 30.0000 1.08324
\(768\) 0 0
\(769\) −37.0000 −1.33425 −0.667127 − 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −24.0000 −0.863779
\(773\) −15.0000 −0.539513 −0.269756 − 0.962929i \(-0.586943\pi\)
−0.269756 + 0.962929i \(0.586943\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 9.00000 0.322666
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) −2.00000 −0.0715656
\(782\) 1.00000 0.0357599
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) −3.00000 −0.106938 −0.0534692 − 0.998569i \(-0.517028\pi\)
−0.0534692 + 0.998569i \(0.517028\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) 60.0000 2.13335
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) −21.0000 −0.745262
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) −31.0000 −1.09808 −0.549038 − 0.835797i \(-0.685006\pi\)
−0.549038 + 0.835797i \(0.685006\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) 0 0
\(802\) 3.00000 0.105934
\(803\) −32.0000 −1.12926
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) −10.0000 −0.351799
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −16.0000 −0.561490
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) −19.0000 −0.664319
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0000 −0.418803 −0.209401 − 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 3.00000 0.104510
\(825\) 0 0
\(826\) −60.0000 −2.08767
\(827\) −38.0000 −1.32139 −0.660695 − 0.750655i \(-0.729738\pi\)
−0.660695 + 0.750655i \(0.729738\pi\)
\(828\) 0 0
\(829\) −16.0000 −0.555703 −0.277851 − 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −9.00000 −0.311832
\(834\) 0 0
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 24.0000 0.829066
\(839\) −27.0000 −0.932144 −0.466072 − 0.884747i \(-0.654331\pi\)
−0.466072 + 0.884747i \(0.654331\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 24.0000 0.827095
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 28.0000 0.962091
\(848\) −13.0000 −0.446422
\(849\) 0 0
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 7.00000 0.239115 0.119558 − 0.992827i \(-0.461852\pi\)
0.119558 + 0.992827i \(0.461852\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −28.0000 −0.953684
\(863\) 40.0000 1.36162 0.680808 − 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 12.0000 0.404980
\(879\) 0 0
\(880\) 0 0
\(881\) −33.0000 −1.11180 −0.555899 − 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 0 0
\(883\) 50.0000 1.68263 0.841317 − 0.540542i \(-0.181781\pi\)
0.841317 + 0.540542i \(0.181781\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) 39.0000 1.31023
\(887\) 15.0000 0.503651 0.251825 − 0.967773i \(-0.418969\pi\)
0.251825 + 0.967773i \(0.418969\pi\)
\(888\) 0 0
\(889\) 64.0000 2.14649
\(890\) 0 0
\(891\) 0 0
\(892\) −17.0000 −0.569202
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −42.0000 −1.40156
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 13.0000 0.433093
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 15.0000 0.498893
\(905\) 0 0
\(906\) 0 0
\(907\) −23.0000 −0.763702 −0.381851 − 0.924224i \(-0.624713\pi\)
−0.381851 + 0.924224i \(0.624713\pi\)
\(908\) −22.0000 −0.730096
\(909\) 0 0
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 − 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) 22.0000 0.728094
\(914\) −31.0000 −1.02539
\(915\) 0 0
\(916\) −18.0000 −0.594737
\(917\) 32.0000 1.05673
\(918\) 0 0
\(919\) −47.0000 −1.55039 −0.775193 − 0.631724i \(-0.782348\pi\)
−0.775193 + 0.631724i \(0.782348\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.00000 −0.0329332
\(923\) −2.00000 −0.0658308
\(924\) 0 0
\(925\) 0 0
\(926\) 35.0000 1.15017
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) 15.0000 0.492134 0.246067 − 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 0 0
\(931\) 72.0000 2.35970
\(932\) 3.00000 0.0982683
\(933\) 0 0
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) 0 0
\(937\) 17.0000 0.555366 0.277683 − 0.960673i \(-0.410434\pi\)
0.277683 + 0.960673i \(0.410434\pi\)
\(938\) 40.0000 1.30605
\(939\) 0 0
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 − 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 0 0
\(943\) 1.00000 0.0325645
\(944\) −15.0000 −0.488208
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 16.0000 0.519930 0.259965 − 0.965618i \(-0.416289\pi\)
0.259965 + 0.965618i \(0.416289\pi\)
\(948\) 0 0
\(949\) −32.0000 −1.03876
\(950\) 0 0
\(951\) 0 0
\(952\) −4.00000 −0.129641
\(953\) 42.0000 1.36051 0.680257 − 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −6.00000 −0.193448
\(963\) 0 0
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) −1.00000 −0.0321578 −0.0160789 − 0.999871i \(-0.505118\pi\)
−0.0160789 + 0.999871i \(0.505118\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) −25.0000 −0.802288 −0.401144 − 0.916015i \(-0.631387\pi\)
−0.401144 + 0.916015i \(0.631387\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) 0 0
\(982\) 39.0000 1.24454
\(983\) −8.00000 −0.255160 −0.127580 − 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 − 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 − 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 41.0000 1.29783
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7650.2.a.c.1.1 1
3.2 odd 2 2550.2.a.t.1.1 yes 1
5.4 even 2 7650.2.a.cj.1.1 1
15.2 even 4 2550.2.d.h.2449.2 2
15.8 even 4 2550.2.d.h.2449.1 2
15.14 odd 2 2550.2.a.q.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.q.1.1 ✓ 1 15.14 odd 2
2550.2.a.t.1.1 yes 1 3.2 odd 2
2550.2.d.h.2449.1 2 15.8 even 4
2550.2.d.h.2449.2 2 15.2 even 4
7650.2.a.c.1.1 1 1.1 even 1 trivial
7650.2.a.cj.1.1 1 5.4 even 2